# What is the instantaneous rate of change of #f(x)=1/(x-7 )# at #x=0 #?

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To find the instantaneous rate of change of ( f(x) = \frac{1}{x - 7} ) at ( x = 0 ), we need to calculate the derivative of ( f(x) ) and then evaluate it at ( x = 0 ).

First, find the derivative of ( f(x) ) using the quotient rule:

[ f'(x) = \frac{d}{dx}\left(\frac{1}{x - 7}\right) ]

[ = \frac{-(1) \cdot (x - 7)^{-2} \cdot (1)}{1} ]

[ = -\frac{1}{(x - 7)^2} ]

Now, evaluate ( f'(x) ) at ( x = 0 ):

[ f'(0) = -\frac{1}{(0 - 7)^2} ]

[ = -\frac{1}{49} ]

So, the instantaneous rate of change of ( f(x) ) at ( x = 0 ) is ( -\frac{1}{49} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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